Measure-expansive systems
C. A. Morales
TL;DR
This work generalizes classical expansiveness to a measure-theoretic setting by introducing $\mu$-expansive dynamics on compact metric spaces. It establishes fundamental equivalences via the diagonal in the product system, proves that the set of converging semi-orbits and periodic points have zero $\mu$-measure, and provides a Denjoy-type classification on the circle alongside interval nonexistence results. The paper also demonstrates probabilistic proofs of standard expansive-system results and extends the theory to continuous maps, linking positive $\mu$-expansiveness to generator concepts and to volume-expanding dynamics through entropy-like quantities. Collectively, these results blend measure theory with topological dynamics to broaden the applicability of expansiveness and offer new probabilistic tools for dynamical systems analysis.
Abstract
We call a dynamical system on a measurable metric space {\em measure-expansive} if the probability of two orbits remain close each other for all time is negligible (i.e. zero). We extend results of expansive systems on compact metric spaces to the measure-expansive context. For instance, the measure-expansive homeomorphisms are characterized as those homeomorphisms $f$ for which the diagonal is almost invariant for $f\times f$ with respect to the product measure. In addition, the set of points with converging semi-orbits for such homeomorphisms have measure zero. In particular, the set of periodic orbits for these homeomorphisms is also of measure zero. We also prove that there are no measure-expansive homeomorphisms in the interval and, in the circle, they are the Denjoy ones. As an application we obtain probabilistic proofs of some result of expansive systems. We also present some analogous results for continuous maps.